We showed in [1]} that a pure, measurement-inclusive, SILL dictionary can make effective low-dimensional Koopman models. Between this section and Appendix  we demonstrate that the homogeneous SILL dictionary satisfies uniform finite approximate closure (see Definition REF  in Section REF ). In essence, uniform finite approximate closure guarantees that the dimensionality of the dictionary space does not need to diverge to infinity, while simultaneously approximating the vector field \(F(y)\)  sufficiently well. This property is what makes numerical approximation of the Koopman generator equation possible.
